Gauged Gromov–Witten theory for small spheres

We relate the genus zero gauged Gromov–Witten invariants of a smooth projective variety for sufficiently small area with equivariant Gromov–Witten invariants. As an application we deduce a gauged version of abelianization for Gromov–Witten invariants in the small area chamber. In the symplectic setting, we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map.     

Witten deformation and the equivariant index

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.      

Elliptic genera of homogeneous spin manifolds and theta functions identities

By Witten rigidity theorem and the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, the elliptic genus of a homogeneous spin manifold G/H can be expressed as a sum of theta functions quotients over the Weyl group of G. Consequently, we obtain several classes of combinatorial identities of theta functions.     

Action hamiltonienne d’un tore et formule de localisation en cohomologie équivariante

Using an idea of Witten (see [8]), we give a localization formula in equivariant cohomology in the case of an Hamiltonian torus action on a compact symplectic manifold χ. The integral over χ of an equivariant closed form can be written as an integral over the submanifold of critical points of the square of the moment map.     

On the rigidity theorem for elliptic genera

We give a detailed proof of the rigidity theorem for elliptic genera. Using the Lefschetz fixed point formula we carefully analyze the relation between the characteristic power series defining the elliptic genera and the equivariant elliptic genera. We show that equivariant elliptic genera converge to Jacobi functions which are holomorphic. This implies the rigidity of elliptic genera. Our approach can be easily modified to give a proof of the rigidity theorem for the elliptic genera of level .

     

Natural equivariant transversally elliptic Dirac operators

We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-valued equivariant index coincide with those of an elliptic operator constructed from the original data.     

The fibrewise Leray–Schauder index

The index constructed by Leray and Schauder in 1934 admits generalizations in two directions to infinite-dimensional fixed-point and vector field indices. We present the constructions of fibrewise equivariant indices of both types and illustrate the definitions by applications to the stable homo-topy Fuller index and Seiberg–Witten invariant. Dedicated to the memory of Jean Leray     

Rigidity of higher elliptic genera

We prove some general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes. We also obtain the vanishing of some higher elliptic genera.Both authors are supported in part by NFS     

Quantum Lefschetz hyperplane theorem

The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov–Witten invariants of X and Gromov–Witten invariants of complete intersections Y in X is established. Oblatum 21-IV-2000 & 11-I-2001?Published online: 2 April 2001     

Power operations in elliptic cohomology and representations of loop groups

Part I of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. Part II discusses a relationship between equivariant elliptic cohomology and representations of loop groups. Part III investigates the representation of theoretic considerations which give rise to the power operations discussed in Part I.

     

The truncated Witten genus

In this paper we define and examine the truncated Witten genus. It is defined as the equivariant index of the Dirac operator on the manifold Map(C_p,M) with its natural C_p-action. Here, Map(C_p,M) is the space of maps from the cyclic group of order p into a closed, connected, spin manifold, M. By applying the Atiyah-Singer index theorem we give a topological formula for the truncated Witten genus which is related to the formula for the Witten genus by truncation of the infinite products. We also show that the equivariant index of the Dirac operator on the projective space Map(C_p,ⁿ⁺¹) is closely related to the truncated Witten genus of ⁿ. The spaces Map(C_p,ⁿ⁺¹) define a filtration of the space Map(S¹,ⁿ⁺¹) which has been used to study equivariant objects on the smooth loop space of ⁿ.Acknowledgement This work was mainly done while the author was studying for a Ph.D. at Warwick University under the supervision of J. D. S. Jones. The author would like to thank John Jones for originally suggesting the problem and for his guidance whilst the author was at Warwick, to thank the EPSRC for funding his Ph.D., and to thank the referee for his detailed comments which helped greatly with the writing of this paper.     

Chiral differential operators: Formal loop group actions and associated modules

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of 2-dimensional σ-models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a “formal loop group action” on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said “formal loop group actions” and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of “formal loop spaces”. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus.     

Equivariant functions and integrals of elliptic functions

In this paper, we introduce the theory of equivariant functions by studying their analytic, geometric and algebraic properties. We also determine the necessary and sufficient conditions under which an equivariant form arises from modular forms. This study was motivated by observing examples of functions for which the Schwarzian derivative is a modular form on a discrete group. We also investigate the Fourier expansions of normalized equivariant functions, and a strong emphasis is made on the connections to elliptic functions and their integrals.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Formules de localisation en cohomologie equivariante

In this paper, we develop a method of localization in equivariant cohomology based on the notion of partition of unity cohomology. We apply this method in two cases. In the first case, this method gives a refinement of the localization of Atiyah–Bott and Berline–Vergne (in the frame given by Bismut). After, we consider the Hamiltonian action of a torus, and we realise, following the idea of Witten, the localization on the critical points of the square of the moment map.     

Vanishing of odd dimensional intersection cohomology II

For a variety where a connected linear algebraic group acts with only finitely many orbits, each of which admits an attractive slice, we show that the stratification by orbits is perfect for equivariant intersection cohomology with respect to any equivariant local system. This applies to provide a relationship between the vanishing of the odd dimensional intersection cohomology sheaves and of the odd dimensional global intersection cohomology groups. For example, we show that odd dimensional intersection cohomology sheaves and global intersection cohomology groups vanish for all complex spherical varieties. Received: 25 February 2000 / Accepted: 15 February 2001 / Published online: 23 July 2001     

The Crepant Resolution Conjecture for Three-Dimensional Flags Modulo an Involution

After fixing a nondegenerate bilinear form on a vector space V, we define a ?₂-action on the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov–Witten potential function of [F/?₂] agrees (up to unstable terms) with the genus zero Gromov–Witten potential function of a crepant resolution Y of the quotient scheme F/?₂, after setting a quantum parameter to ?1, making a linear change of variables, and analytically continuing coefficients. We explicitly compute several invariants for the orbifold and the resolution, then argue that these determine the others via basic properties of Gromov–Witten invariants.